Question: Given that $x+\sin y=2008$ and $x+2008 \cos y=2007$, where $0 \leq y \leq \frac{\pi}{2}$, find the value of $x+y$.
Explanation: Subtracting the two equations gives $\sin y - 2008 \cos y = 1$. But since $0 \leq y \leq \frac{\pi}{2}$, the maximum of $\sin y$ is 1 and the minimum of $\cos y$ is 0, so we must have $\sin y = 1$, so $y = \frac{\pi}{2}$ and $x = 2007,$ so $x+y = \boxed{2007 + \frac\pi 2}$.